The Kingdom of the North is an independent kingdom in Westeros currently ruled by King Jon Snow. Following the War of Conquest, the North became one of the constituent regions of the Seven Kingdoms. It has since declared its independence in defiance of the Iron Throne.
The Kingdom of the North was founded during the time of the First Men by Bran the Builder, the first King in the North. At its time of greatest expansion, the Kingdom of the North included all the lands north of the Neck and south of the Gift. Bear Island became part of the Kingdom of the North after King Rodrik Stark won it from the ironborn and the rule of House Hoare in, according to legend, a wrestling match.
The Kingdom of the North was the only kingdom of the First Men that successfully resisted the Andal Invasion and clung to its ways and religion. Southern kingdoms fell and some dynasties were replaced by Andal ones and others were simply absorbed despite family names remaining the same - as was the case of the ancient House Gardener.
Eventually, it was absorbed into the unified monarchy of House Targaryen after King Torrhen Stark bent the knee to Aegon the Conqueror to spare his people from the fate suffered by the armies of the Kingdom of the Rock and Kingdom of the Reach in the Field of Fire.
The Kingdom of the North, however, rose again as one of the consequences of the Execution of Eddard Stark when Robb Stark, Eddard's eldest son and heir, is declared King in the North by his own vassals, as well as the Riverlords sworn to his maternal grandfather, Hoster Tully, the Lord Paramount of the Trident. Thus, the new Kingdom of the North included all lands historically ruled by House Stark, and the lands surrounding the Trident, ruled by House Tully since the times of Aegon's Conquest.
During the beginning of the War of the Five Kings, the Kingdom of the North held full control over the lands of the the North and the the Riverlands. After destroying the Lannister forces in the Riverlands, King Robb Stark launched a limited invasion of the Westerlands, in order to draw Tywin Lannister out of Harrenhal and finish off his remaining army.
This revived Kingdom of the North would only last for two years when King Robb Stark and his entire army were killed and massacred at the Red Wedding orchestrated by the Lannisters, along with the Boltons and Freys. After a brief few years of Bolton rule, who ruled the North on behalf of the Iron Throne in the aftermath of the Red Wedding, House Bolton was narrowly overthrown by the surviving forces under House Stark in the Battle of the Bastards due to Sansa Stark summoning the knights of the Vale sworn to her maternal cousin Robin Arryn, the Lord Paramount of the Vale.
Jon Snow, believed to be Lord Eddard's last living son (but in reality his nephew) after the death of his alleged younger half-brother Rickon Stark, was proclaimed the new King in the North by the Northern lords, despite his bastard status. He also has the support of the Free Folk, whom he helped save at Hardhome, and the Vale lords (whose liege lord, Robin Arryn, may secretly share blood with Jon as well as the other Stark children, albeit much more distant). It is unknown, but perhaps likely, that various houses of the Riverlands, such as House Mallister and House Blackwood, may declare for Jon Snow and the new Kingdom of the North, as they were loyal bannermen to House Tully. Furthermore, these houses are said to be in open rebellion against House Frey, even though the current head of House Tully, Lord Edmure Tully, remains their prisoner. With the deaths of Walder Frey and his two sons, Lothar Frey and Black Walder, at the hands of Arya Stark at the Twins, these houses may be more inclined to once again declare their independence and rejoin the North, especially now that they are joined by the Vale of Arryn, whose ruler, Robin Arryn, as well as King Jon's surviving alleged half-siblings - Sansa, Arya, and Bran - share blood with House Tully through their mothers, Lysa and Catelyn.
Yet Bran Stark, Lord Eddard's last living trueborn son, has learned through his gift of the greensight, that King Jon is actually his cousin through his aunt, Lyanna Stark. Jon's true father was the Prince of Dragonstone, Rhaegar Targaryen, whose alleged kidnapping of Lyanna along with the subsequent execution of Rickard and Brandon Stark, his maternal grandfather and uncle, were both the catalyst for Houses Stark, Arryn, and Tully uniting with House Baratheon to overthrow the Targaryen dynasty in Robert's Rebellion.
A passage in High Septon Maynard's meticulous records — discovered by Gilly in scrolls taken by Samwell Tarly from the Citadel — says that Maynard annulled the marriage of Rhaegar and Elia Martell and then officiated the secret wedding of Rhaegar and Lyanna in Dorne, where Jon was later born. This would make Jon the trueborn head of House Targaryen instead of the current claimant Daenerys Targaryen, his paternal aunt. Although Jon has Stark blood from his mother's side, he is not the rightful head of House Stark.
Between the threat of the Night King and his army of White Walkers and wights and the impending arrival of Daenerys Targaryen and her dragons, what course of action, if any, this new Kingdom of the North will take to stay independent remains to be seen.